Research Activities
In the Laboratory of Evolution Equations investigations in the following main directions are carried on:
General
theory of initial boundary value problems for linear
equations and systems
of elliptic, parabolic, and quasi-elliptic types,
including the questions of existence of classical
solutions,
solutions estimating in uniform norms,
boundary value problems in domains with a nonsmooth
boundary.
Theory
of nonlinear evolutionary equations of parabolic and
hyperbolic types,
questions of solvability of basic initial boundary value
problems
“on the whole” with respect to time,
asymptotic behavior of solutions, their stabilization and
stability,
attraction domains of stable stationary or periodic
solutions,
theorems on integral manifolds.
Lyapunov
methods in the problem of stability of solutions
to linearized parabolic and hyperbolic systems,
localization of the spectrum of corresponding
differential operators,
and solvability of hereby arising operator equations.
Application
of the spectral theory of differential operators to
solving inverse problems
for evolutionary equations of mathematical physics.
General
questions of calculus of variations and its applications
to solution
of problems of nonlinear theory of elasticity.
Spectral
properties of operators arising in the theory
of small oscillations of rotating ideal fluid
and dependence of qualitative characteristic features of
solutions
on the character of the spectrum of the corresponding
problem.